Category Archives: Maths

On July 20 1969 The Apollo 11 Lunar Module touched down on the surface of the moon and Neil Armstrong and Buzz Aldrin began their walk. Many (including me) judge this to be mankind’s greatest single-event achievement so far. Outlined below are the many aspects of this story which provide learning opportunities and potential exam questions across the three GCSE Sciences, particularly Physics.

The take-off

Despite the enormous sound and visual fury of the launch, the fuel used by the Saturn rockets powering the mission was mainly not fossil fuel, rather it was a mixture of liquid oxygen and hydrogen. Normally gaseous, very low temperatures are required to liquefy them, -219 C and -253 C respectively. Being liquid rather than gas is safer, and occupies much less space because volume = mass / density and liquid density is higher. Saturn had sections which as fuel was used up were jettisoned to just leave the lunar and command modules. The enormous power was needed to enable the modules to reach the required speed to exit the earth’s atmosphere and escape the main gravitational pull.

Saturn V launches Apollo 11

The journey there

The distance from the earth to the moon is about 240,000 miles and the maximum speed was just over 24,000 miles per hour as it left earth’s orbit. So a “time = distance / speed “ calculation indicates a ten hour journey time and yet it took 3 days, so what happened to this slow-coach! Well, maximum speed does not mean average speed, and after the Saturn rockets were jettisoned, gravity slowed down the un-powered Module , as required, in order not to fly straight past the moon as it approached. Also, the journey included an orbit of the earth and several of the moon before descending to the moon so the distance was much higher.

The journey there..and back

Fuel Cells

If the rockets were jettisoned, how did the modules get to the moon without their powerful fuel? Well, once the modules were propelled out of earth’s orbit at high speed, less force was acting upon them since air resistance was zero. There was still a backwards gravitational pull of earth but it became smaller and smaller. So Newton’s Law would suggest they just carry on in the direction they were pointing, namely towards the moon, even without Saturn rockets’s major fuel source, albeit gradually decelerating from initial 24,000 mph. Small amounts of fuel were needed for lighting, communication and landing/leaving the moon, and these were a mixture of conventional fuels and fuel cells developed in Cambridge University, which were the early versions of the fuel cells we learn about in Physics GCSE. Namely hydrogen plus oxygen combining through electrodes to produce water,and release energy as electricity. The maximum power was around 2000 Watts and the water was not wasted – it was drunk by the astronauts!

A fuel cell was used on Apollo 11 mission

“In space, no one can hear you scream”

As the advert for the science fiction classic confirmed, sound cannot travel in space because the longitudinal sound waves, whose vibrations are parallel to the direction of travel, need particles such as air to vibrate – but there is no air in a vacuum. So how come we could hear the astronauts?

Sound waves don’t travel in space …but radio waves do

As David Bowie memorably told us in Space Oddity, a conversation was possible between Major Tom and Ground Control. Well, the answer is that communication was achieved by Radio waves, which are not sound waves but Electromagnetic waves which as transverse waves vibrate at right angles to the direction of travel. Just like other parts of the spectrum – like light waves from the sun – radio waves can travel through a vacuum at the speed of light namely 300 million meters per second. Since the 240,000 miles is around 360 million meters, then using time = distance / speed, the time for a radio signal to travel from the moon to the earth is only 1.2 seconds. Hence the only-slight delay between Houston asking a question and the astronauts answering.

Note however that Michael Collins, alone in the Command module while Armstrong and Aldrin walked on the moon, could not be contacted on the far side of the moon as radio contact was lost, as expected. Perhaps this was why Bowie’s Major Tom lost contact at the end of the record – “can you hear me Major Tom?”

Ground Control could hear Major Tom …at first

The physics of an orbit

When the lunar module had jettisoned its rockets it performed an orbit of the earth before heading to the moon. How does this work? If the module is set in forward motion at just the right speed then the force at right angles to its motion – namely gravity – pulls it towards earth and the net result is a bisecting direction along the path of the orbit.

The speed of the orbit remains constant at 25,000 miles an hour but the velocity is constantly changing. How can this be? Well, it’s because velocity is a vector and speed is a scalar quantity and as Vector tells Gru in Despicable Me, a vector has magnitude as well as direction. So the velocity is constantly changing because the direction in a circular path is constantly changing. When a force creates a circular motion, this is a centripetal force. (Gravity is a non-contact force while other centripetal forces are contact forces – the friction when a motor bike turns, and the tension in the spokes of the London Eye)

The diameter of the earth is about 8000 miles and the Module initially orbited the earth at around 100 miles up. So the diameter of the orbit around the centre of the earth was 8200 miles, giving a circumference of approximately 25,000 miles using Pi. At almost 25,000 miles per hour, the initial orbit took 1 hour.

The Moon Landing

When Armstrong and Aldrin’s lunar module separated from Collins’s Command Module above the moon, it reduced its speed but slightly overshot the landing site in the Sea of Tranquility in order to avoid landing in a crater. Armstrong took over control from the Module computer to achieve this ( a computer with less processing power than an I Phone incidentally). Less than 30 seconds of fuel remained, so this was where both of the astronauts’ flying experience, including dog fights with Russian MIG’s in the Korean War, proved invaluable. They stayed impossibly cool, while Houston’s control centre personnel famously were so tense they almost “turned blue”.

Armstrong’s heart beat stayed normal at 70 beats per minute, almost until the “Eagle has landed” but even he succumbed at touchdown to the fight or flight adrenaline hormone at touch down, when his heartbeat reached 150.

one small step for man….

After Armstrong stepped down off the ladder – “one small step for man, one giant leap for mankind” – Aldrin soon followed him and began, as the Police would later sing, while Walking on the Moon, to take “giant steps” with his “feet hardly touching the ground”. Why is this? Well ,gravity there is only a sixth of the earth’s gravity ( g is 1.6 rather than 10). So it was easy to hop around. And why is the gravitational force lower? Because the force is proportional to the mass of the two objects, and the moon is lighter than the earth, even if the man has the same mass. So a person of 50 kg faces a gravitational downward force of 500 N on earth but only 80 N on the moon.

The Police …drums on the side of a Saturn

They collected rocks and when later analysed they were found
to contain the chemical lelements silicon, iron, aluminum, calcium, magnesium, titanium and oxygen. No
carbon or nitrogen, so not enough ingredients for biological life. Years
later however , hydrogen and iced water were found at the moon’s poles and this
opens the possibility, with the presence of hydrogen and oxygen, of creating
fuel cells using electrolysis which could mean that the Moon could be used as
refuelling stop on the way to Mars.

The journey back

After taking off from the moon, the lunar module docked with the orbiting Command Module and together they returned to earth. Long before the mission, Aldrin had written a thesis on docking in space based on his experience as a scientist and Air Force pilot in Korea. As the Module approached the earth atmosphere the frictional force – this time a contact force – caused the heat shield to reach high temperatures and gradually melt – as planned.

Splashdown

A parachute slowed the Module down further, with air resistance offsetting the weight of the Module, which floated down at a leisurely terminal velocity to the sea.

The crew were kept in quarantine for several days in case they had caught viruses on the moon. A virus – unlike bacteria – is counted as non-living but nevertheless can contain DNA. It is worth recalling that DNA was discovered by Watson and Crick at Cambridge University only 16 years before the Apollo 11 mission.

Further Physics
work

All of the above science should be readily understandable by anyone taking Physics or Maths GCSE – if not it’s a definite revision topic! For those carrying on with Physics, the A Level and Physics Aptitude Test for Oxford will contain more advanced Space concepts like eclipses, Kepler’s Laws for orbits and what many consider to be one of the all-time great equations; namely Newton’s formula for the Force exerted by gravity on two objects, of mass m1 and m2: F = Gm1m2/r^2 where r is the distance between the masses and G is the universal gravitational constant.

Scientists are still not sure what Gravity truly is, yet in the 1700’s Newton could already quantify it, and in a sense invented the science behind Apollo.

I have just completed some Maths tutoring for two excellent students hoping to join a grammar or independent school in South West London. Their approach was exemplary, their Maths was already well in advance of Year 6, and they wanted to get even better, being prepared to work very hard in lessons and at home. One full practice paper was not enough for homework, they coped with two a week. Their parents hoped for a free or reduced fees place, but if not I have no doubt they would try to find a way to sacrifice to pay fees.

With the recent news about possible expansion of grammar schools, it made me think about what would happen if my two students did, or didn’t, make the grammar schools, and also how the various entry exams compared to each other, and to traditional year 6 SATs standards. In other words, what should pupils expect in their exam? Let’s start with this.

The entrance exam

My focus was upon my local South West London schools, 10 fee paying private independent schools and 8 free, state, selective grammar schools. I drew broad conclusions about the latest exam processes, likely to be reasonably applicable outside London too. The first thing to say is that in these 18 Schools, it is very difficult to find free sample papers or even sample questions on their websites. This is to avoid advantaged children “buying” their entrance through expensive “teaching to the test” tuition. However, for some of the Surrey schools typical common entrance papers can be purchased, some schools just outside this area do publish sample papers, and of course national publishers like CGP and Bond make practice papers available.

So you can piece together what the typical test will look like. Maths rather than English is my speciality so here are some of features of the typical Maths entrance paper.

The number of questions will be between 25 and 50, students have 45 minutes to 75 minutes to complete, so at 1.5 to 2 minutes each these are short sharp questions. But the complexity varies significantly from beginning to end, so you should expect to spend 30 seconds on the easy ones and perhaps 3 minutes on the difficult ones. The ability to work fast is almost as important as the ability to answer the question. The paper typically divides, in order of questions, into what I’ll call the four quartiles of difficulty. Remember that the higher the reputation of the school, the higher the demand for places, the higher proportion of questions in quartiles 3 and 4, as follows:

4th quartile – Beyond KS2 to KS3 and KS4 GCSE, and PuzzlesNumber : Exchange rate conversions, Fibonacci sequence, Prime factor trees, Ratio problems such as cake recipe; HCF and LCM; powers.Algebra: simultaneous equations created from e.g. prices of burgers and soft drinks, Multiply double brackets using grid or FOILGeometry Parallel line angles, enlargements and scale factors, 3-D cuboidsData: Venn diagrams, Probability, Mode, Range and MedianProblems: Sudoku-like magic number puzzles, Shapes representing operations, number machines Shortest route problems such as through the streets of New York; full page multi-paragraph problems featuring combination of numeric and verbal reason logic culminating in for example, which of five children got a present, which of five animal lives on which island?
This last, 4th quartile frequently goes well beyond KS2 in two respects. Firstly, what I’ll call “puzzles” – which ironically will never resurface in secondary exams. Secondly, school syllabus content stretching well into KS3 Year 8 and 9 even in rare cases up to KS4 GCSE level (yes!). This last quartile contains the differentiator questions, the ones you have to be able to do to be really confident of gaining entry. When tutoring an 11 plus pupil followed by a GCSE pupil I sometimes find myself using the same sample questions.

Most of the school websites say, to be politically correct, that the questions should be suitable for any KS2 student (only one admitted that some questions may stretch to KS3). Parents should not be fooled. With demand outstripping supply by 4 or more to 1, the higher reputation schools do throw in the puzzles and year 7-11 level questions to identify the brightest pupils.

How many exams?

The table below shows, for the 8 grammar schools sampled in SW London, most have 2 stages, although the Sutton set start with the common SET test. The 10 independents all have just one stage except St Pauls, which starts with the common ISBE test, and most have an interview to confirm selection. Most schools feature Maths, English and either a separate verbal reasoning test or similar questions within English. What is noticeable is that Non Verbal Reasoning is becoming quite rare now (thank goodness – awfully difficult to teach!)

School
(State selective)

Maths

English

Non Verbal
Reasoning

Comments

Sample Maths available

Tiffin Boys

Y

Y

N

2 stages, and stage 1 counts 10% each Maths and English, Stage 2 counts 40% each Maths and English for entry.

No

Tiffin Girls

Y

Y

N

2 stages, and stage 1 is Maths and English OMR multi choice, passing gets you to Stage 2 Maths and English which alone determines entry

No

Below are the Sutton Grammars taking common SET

Nonsuch High for
Girls

Y

Y

N

2 stage, 1st English and Maths common SET multi choice, then joint second stage Maths and English with Wallington High School for Girls

No but SET samples can be ordered

Wallington
High for Girls

Y

Y

N

2 stage, 1st English and Maths common SET multi choice then joint second stage Maths and English with NonSuch High School for Girls

As above

Greenshaw High

Y

Y

N

1 stage only Maths and English common SET multi choice. Pass for eligibility for 60 places.

As above

Sutton Boys

Y

Y

N

2 stages, first is common SET English and Maths multi choice , to get you to second stage Sutton specific English and Maths. 1st and 2nd stage tests all affect final entry, ratio is 2:2:3:3

As above

Wallington County

Y

Y

N

1 stage only, Maths and English common SET, pass to be eligible for place

As above

Wilson’s Sutton

Y

Y

N

2 Stage , first is common SET Maths and English, second Maths and English. Count in ratio 2:4:4.

Through the lens of my two students, if they started Year 7 even in the best of the local state schools, they would be so far ahead that they would, to be honest, be bored and held back. Like many bright children they need the challenge. The supply of free grammar schools is limited. At many of our local grammars the ratio of applicants to places is 4 to 1 and at some even higher, where queues around the block form at the start of exam day. (Some now phase exams through the day to avoid this). In business, if supply is limited and demand is high, you increase prices or create more capacity – in this case by creating more grammar schools, because prices are fixed at zero.

However the downside is of course that if the brightest pupils are creamed off from state schools, the overall standard must surely fall. This is detrimental to the remaining pupils, who lose the chance to learn from the approach and abilities of high achieving pupils, and dispiriting for teachers who enjoy challenging them and getting a positive can-do response. Some teachers would surely jump ship. Some Headteachers have said this would recreate “secondary moderns”.

One compromise – which one of our local state schools already employs – is to offer a limited number of exam-selective places, while mainly offering free places for local pupils. The question then is, do you sprinkle the selected pupils among the classes, or “set” from the start. The problem with the first approach is that schools are constrained by the national curriculum which prescribes certain content for certain years, so the brighter pupils would be constrained by the pace of the slowest. The alternative is to “Set” from year 7 and effectively teach the top set Year 8 or 9 level content from age 11, and take all GCSE’s (not just Maths) a year early. This “grammar stream” approach is advocated by former UCAS Chief Executive Mary Curnock Cook Or go further (as my old school used to do) and identify the brightest year 7 pupils and to remove them at year end from Year 8 and place them straight into Year 9 (we were called “removes”).

Is tutoring needed?As noted above, the questions definitely stretch beyond standard KS2 (whatever schools say). The question is, how do you get access to, and practice these. In theory, purchase of Bond or CGP practice books can do the trick, but the risk is that the pupil will miss the personal explanation and without homework being set, may not practice enough, and even these excellent publications don’t include the outrageously tricky questions which do crop up. Note also that while common entrance papers like SET the Selective Eligibility Test can be purchased, frequently these are only for Stage 1 permission to sit the really challenging Stage 2 papers which are not formally available. So structured learning, and exam tips are needed over and above school provision. Parents might provide this but many would struggle with the vital end of paper questions. Extra tutoring is your insurance policy (but not a guarantee) and this can come in several forms, including private one to one, or exam centre cramming.

What is tutoring providing?

What you are trying to do is this: First make sure the basics of KS2 are in place. Second, introduce the pupil to a selection of KS3 topics which may crop up. Third, help the pupil work at speed. Fourth, teach exam techniques. Finally set a sufficient quantity and quality of challenging tasks from which gradual improvement instils confidence – the “more I practice the luckier I get”. What is difficult to teach is the natural mathematical abilities such as puzzle solving and spatial awareness, and my guess is that is why such puzzles are included – there may be disadvantaged pupils who cannot afford tutoring yet have that innate mathematical ability which money can’t buy.

In conclusion

The 11 plus is highly challenging. A good KS2 performance – an 11 plus “pass” – will probably not be enough to get through. There are many pupils and parents willing to take up that challenge, to achieve that extra level of excellence. Schools, the State and Tutors all have a part to play in meeting that demand.

Initially to be piloted in around half of our primary schools, the technique involves learning techniques more by rote, asking one child to answer a question, then asking the remainder of children to repeat the answer. The class does not move on until all the class has “got it”. The brighter children avoid being held back because they have a role in leading the other children with the first answer. There are some similarities with Kumon, namely keep practising by repetition until “mastery” of a topic is achieved to an advanced level, but differences too: Chinese Maths emphasises the role of the respected teacher at the front of class, Kumon relies more on self learning through worksheets.

Chinsese children themselves are believed to be 2-3 years ahead by the time they move to Year 11; so 16 year olds in China are already at the same level of maths as an 18 year old A level student in the UK.

There is a view that culturally some British pupils are not ready for this and our cultural diversity and child centred participation doesn’t sit easily with chanting and learning by rote which is common and part of the educational ethos in China. The benefits are not at all questioned.

Chinese Maths versus English real world approach

More important I believe is this. The direction in Maths and Science in England is to introduce more “real world” relevance to exam questions, not just at GCSE KS4 but also at earlier KS3 and KS2 as well.

So while introducing a “back to basics” learning approach in Maths is very good, not least because we are slipping down the international educational league tables, I wonder if joined up thinking is taking place in Government in terms of the following two factors:

If teaching methods move in the direction of focusing upon purely numerical excellence, and yet examiners insist on setting real world applied questions where the maths technique is merely a small means to an end, do we risk the recent Biology GCSE “drunken rat” exam problem ? By this I mean that the children aim to learn the syllabus and technical methods to the best of their ability, and they put a lot of effort into mastering the knowledge and technique required in the syllabus, but meanwhile the examiners smother the questions in “real-world” unfathomable words and situations. So the child learns the techniques but can’t do the exam questions because they haven’t been schooled in the methods of deciphering them, or applying the technical knwledge they have acquired.

An example in Maths itself is the 2015 GCSE question that went viral. The question involved two techniques rarely seen together: algebra, and probability. One can imagine pupils achieving high levels in these two topics individually using Chinese techniques of practising lots of examples, but being unbable to piece together the required jigsaw which requires a different sort of skill altogether.

Mile long and centimeter deep

One other phrase associated with Chinese Maths is interesting: their criticism of the British Maths syallabus is that it is a “mile long but a centimetre deep”. There is something in this. For GCSE Maths there are five basic topics such as Number and Algebra but within those there are many sub-topics making around 80 in all. One wonders if all of these are necessary, for instance frequency density histograms are beloved by specification setters but in practice are never used by businesses. Could some topics be left out allowing time for in-depth understanding of the core?

But we are where we are: my philosophy as a tutor is to “teach to the test”, whether GCSE exams or earlier end-term tests. Because that’s what parents want. And the last thing a child wants is to open an exam paper and find there are topics they don’t even recognise. So you have to teach the whole syallabus, not just the mathematic principles but the ability to understand and answer increasingly inscrutable questions.

Measuring success

In summary there will almost certainly be benefits and we need somehow to catch up on global competitors. An intangible benefit may be a cultural change, to make Maths excellence expected rather than optional. But ultimately, the acid test is this: will the programme lead to better GCSE results, either higher marks, or the same marks at a younger age? This may depend on whether the new techniques are compatible with the direction of Maths exam question designers. Sound learning of fundamentals is essential and surely must be improved – so we have to start somewhere; but it may be only the first base-camp stage in achieving the summit of maths mastery. We may not be able to judge success for half a decade.

In GCSE Science and Maths you are often asked to draw or interpret graphs – representing and visualising data are the technical terms. Often it depends on whether the data is discreet or continuous.

Continuous data can be almost anything – a temperature measurement for instance – and line graphs are generaly used – whereas discreet tends to be categories that can only have certain values and bar charts are best. As an example here is an assessement I did for my hobby – assembled the top 3 pop singles each year for the last 60 years. I used a bar chart to show which artists had appeared more than twice. Not suprisingly the Beatles, Elvis and Michael Jackson were at the top. If you are pop rock and soul fan you can see the full list and how they they were chosen in this link.

As part of my tuition I run through each of the types of graphs you can see here including scatter, line, Pie, box plot, bar, cumulative frequency, histogram. These are becoming ever more important to understand with the new GCSE’s coming next year with Maths.

Another favourite with the examiners expecially with science is the concept of independent and dependent variables. Independent variables are the things you change deliberately e.g. the size of the pellets in a chemcial reaction, and these normally go on the x-axis. These “cause” a change in the dependent variables which are the “effect” i.e.they tend to be continuous, could be the reaction rate, and are usually on the y axis of a graph. Finally the “control variable” is something you keep the same to be fair, such as as room temperature or weight of pellets.

There is often a cross over between Maths and Physics so if you learn about Distance Time graphs in Maths you will also see efectively the same graph in Physics.

And often you will be asked to interpret a graph about which you know nothing such as the drunkne rat biology question – the key is not to panic and instead apply the pronciples you have learned about graph interpretaton.

News today that the Department for Education inadvertently but helpfully posted a SATS test on a practice paper website some time before the real thing (not a good week for the department, the National Audit Office found holes in their accounts). This made me think, how much would it really help to see an exam before?

The answer is, if you didn’t know it was going to be the real thing, then it wouldnt help that much. It would be easy to forget the solutions, especially if time passed.

However, if you did know it was going to be the exam, you would take extra care to remember the methods and solutions.

In practice the chances of this happening are almost zero. Or are they? In the following sense this does happen.

Certain questions occur time after time in pretty much the same form – just with different numbers. Actually this tends to happen more in A-Level than GCSE, but consider these examples:

They are in effect the same question, same technique, but with different numbers.

Will esentially the same question occur in 2016? We shall find out soon. You could look at it in two ways. Either, it occurs so often it’s time for a break: or it’s a staple question, it will occur agan. Second guessing the examiner’s mind is impossible in terms of exact questions, but broadly you can predict the type of question.

What’s clear is that this type of algebra, whether “expand the brackets”, or perhaps the reverse – “factorise”, introduce the brackets, and solve the quadratic – is likely to crop up.

Therefore if you have done your past paper practice, and it does reappear, then in effect you have seen the question before. At least the method, which you have practiced and mastered. If you turn over the paper and see this type of question, you think “joy, I know how to do this”.

Of course not every question is a “repeat” question, but broadly quite a large proportion have similarities. As a back up to learning the methods, past paper practice, with access to worked answers, is so incredibly useful ! And why my tutoring homework always includes some real (on paper, not PC ) past paper examples.

The start of the new English cricket season reminds me of that moment a few weeks ago when England were about to win the T20 World Cup. West Indies batsmen had a mountain to climb from their last over and Ben Stokes, England’s expert “death” bowler, was ready (Stokes had won the game for England in the semi final in similar circumstances). However (and this is a personal opinion) I think Stokes was already imagining his celebration, particularly to his nemesis Marlon Samuels, and his concentration wavered.

Carlos Brathwaite, West Indian batsman, had other ideas and England’s hopes disappeared in a blaze of sixes. His achievement lends itself to a form of GCSE Maths question proving popular with examiners. Namely the “reverse mean” question where pupils have to calculate not the average of a given set of numbers, rather the number needed to change an old mean to a new mean.

England cricket team had an average of 7.75 runs from their 20 overs. West Indies after 19 overs had averaged 7.21 runs per over. How many runs did they need from the last over to win the match ? (i.e. exceed England’s total by 1 run) (See below for answer)

Although West Indian cricket has struggled of late, the win was eventually going to happen as they have a tremendous competitive spirit. As can be see in this fascinating BBC article about the use of gamification in Jamacan classes. A small company, Edufocal, has set up computer aided classrooms for core subjects like Maths which reward the children for scoring right answers. The company is growing and results are improving. Sponsorship from Virgin’s Branson Centre of Entrepreurship is helping. Some of my pupils use CGP Mathsbuster which has a similar philosophy – bronze, silver, gold trophies are awarded as pupils move through the questions. But in Edufocal’s case the prizes are real – cinema tickets etc ( funded by subscription).

And just to keep the Jamaican theme going, one of my favourite artists and songs is Bob Marley’s One Love, the video here being not in Jamaica but London, with a yound Suggs and Paul McCartney.

So, returning to the cricket, the final was featured in the “Cricinfo” website which specialises in statistical coverage of cricket matches and includes graphs which, while not exactly the same as in GCSE, do show the power of using visual techniques to bring numbers to life.

Carlos Brathwaite perhaps wasn’t thinking of solving a GCSE puzzle as he awaited Stokes’s first ball (a glance at the scoreboard may have been easier!) But if he was. here is what he would be calculating:

England’s average (mean) of 7.75 runs per over from 20 overs meant they scored 7.75 x 20 =155 runs in total. So West Indies needed to score 156 to win. But after 19 overs, their average was only 7.21 per over so they had scored 7.21 x 19 = 137 runs. So from the last over they needed 156 minus 137 equals 19 runs to win. (In fact they scored 3 sixes from 3 balls, and another from the next for good measure, to win with 2 balls left. Bravo!)

Big data is a term used increasingly to describe the use of large amounts of data gathered electronically to determine insights otherwise lost in the detail. It is often characterised as the 3 V’s, Volume ( e.g. terrabytes); Variety (e.g. social media insights as well as surveys) and Velocity (fast data transfer and processing). A famous early example was use of location-specific Google searches on flu medicine to predict and track the spread of a flu virus through America quicker than conventional methids.

How can this principle help revision? Well, a subset of Big Data is simply “more data than usual” – big data light to coin a phrase – and I have done this with Maths past papers. Not just answers and methods are available on line for all past papers – that is well known – but also examiners’ comments are available question by question.

Initially I have looked in detail at 4 past papers, 100 questions, and captured the comments from each, 130 in all, for which the examiners highlighted common causes of lost marks from tens of thousands of entries. Then I grouped them and tallied them GCSE style in a frequency table and chart as seen left. This sheds light on general areas for revision, with (lack of) “basic maths skills” ferquently bemoaned by examiners, as well as subtle tips such as “read the question very carefully and make sure you show working”.

Then, further I picked out twenty very specific examples of errors that seemed to occur – this time syallabus technical content rather than functional categories above – and wrote and illustrated a list of “20 things examiners do and don’t like to see”. A typical one is shown. I recently took some examiner marking training and I can assure you this is true. If a question asks you to “express a number as a product of its prime factors” then merely listing them, with commas, will lose you a mark, if the “times” sign is missed out. Even if the numbers are riight as above.

I have put these and other tips together into an Easter/Summer Term special lesson covering:
– reminder of key basic maths skills, especially the ones that get overlooked
– exam technique : start, during and end of exam
– problem solving techniques for difficult, wordy, end of paper high mark questions
– active revision methods
– the twenty things examiners do and don’t want to see.

With half time biscuits covering the Jack Black “school of rock” Math video.

Another phrase used in the Big Data field is “wisdom of the crowds”. This is being applied superbly by the excellent on-line Maths tutor “MrBartonMaths” (he is also a real teacher). One of the blog pages he runs is “Diagnostic Questions – Guess the Misconception” where students are invited on-line to answer a multiple choice question and give their reason (a typical weekly question is shown). Typically around a thousand students vote (hence the “crowd”) and reveal what errors are often being made (in the example A is correct of course, C was the most common error). The misconceptions are both the students’ errors, and tutors’ sometimes incorrect expectations of what errors might be most frequent.

A lot of data is available out there on-line – the key is to process and present it in the best way to understand and hence help students.

News that Psy’s worldwide hit “Gangnam Style” has exceeded 2.5 billion video views is astonishing. That’s two thousand five hundred million (like when my football team loses 8-0, the teleprinter helpfully adds “eight”). A Maths GCSE question could be:

Write 2,500,000,000 in standard form : Ans. 2.5 x 10 to the 9th

But where or what is Gangnam? Well, the Economist reported recently from Gangnam itself in Seoul, South Korea, where just to get into the best private tuition after-school study groups, children have to pass exams; the children are cramming for crammers. These are the Hagwon schools and the best are called Sekki (cub) – most of them in fashionable Daechi-dong in stylish Gangnam (yes that one). Students work at a level up to 5 years ahead of their age group syllabus and often arrive home tired and late after a double day in education. A law is now being proposed to ban children from studying in private tuition after 10 pm.

Children also spend their free periods at school doing extra homework for Hagwon. Parents spend 0.8 % of GDP (or a tenth of all household income) on private education, which puts South Korea top on the Private Tuition World League (Britain is 8th with 0.4% of GDP). But few parents actually admit to enrolling.

But this is what we in the West are up against – huge achievement in South East Asia. Demand for tuition is so high (sigh!) in Seoul, South Korea that no advertising is needed.

But does this have a measurable impact upon results? Well, yes. according to the latest PISA study (not the leaning tower, rather the international education benchmark for 15 year olds in 72 countries). Korea is in the top ten for Maths and reading and 11th for Science (Singapore as ever dominates). While the U.K. has climbed to 15th in Science it has dropped to 27th in Maths. A sobering thought. Should the U.K. strive to match SE Asia by copying their “learning by wrote” mastery techniques, or push on with our strategy of “real world” syllabus questions perhaps more relevant to the workplace. That’s for a future blog!

Gangnam is a fashionable district of Seoul in South Korea described as affluent and the equivalent of Beverley Hills or Chelsea. Psy wrote “Gangnam Style” as a slightly ironic social comment on Gangnam residents lifestyle.

News perhaps lost over Christmas was that national tests are to be introduced by the Goverment for times tables. up to times 12 by age 11. Momentous not so much for the fact that “3R’s back to basics” are being tested – it seems to makes sense to do so – but for the first time ever a national test is to be conducted on-line with results available immediately. It is another test for teachers to organise, so more workload, but hopefully the automation minimises administration and marking (provided the iT works !)

No doubt someone will beaver away analysing where the hotspots and coldspots are ( will x7 prove the most difficult, except in Sevenoaks? Will x2 prove the easiest, especially in Twice Brewed?). A benefit of “Big Data” analysis is that it reveals “Wisdom of the Crowds”, or “Bulk Crime” as Police would call it, where when you are able to easily consolidate data, patterns emerge, which can lead to actions being addressed.

We are all getting used to using on-line Maths coaching and testing, there are scores of websites. My own favourites are CGP Mathsbuster, BBC Bitesize, AQA AllAboutMaths, http://www.cimt.plymouth.ac.uk/ .

And last but not least http://www.mrbartonmaths.com/ where he uses “Essential Skills” diagnostic Maths quizzes with the ingenious requirement to add a few lines on “why you believe the answer is correct”, which on compilation reveals the top reasons why pupils get a particular question wrong e.g in BIDMAS. And so “Wisdom of the Crowds” helps tutors and teachers identify problem areas with the certain knowledge that a large number of other pupils also find a topic difficult.

In conclusion, how relevent is the story for GCSE? Well, the national on line test is another step on the road to automation (how far will it go?) and while Times tables will clearly not be asked directly in GCSE, many steps in GCSE questions do require a thorough knowledge of the basics, especially the non-calculator exam, otherwise slow or incorrect answers will result.

Athletics has had a bad press recently, rightly so. But let’s celebrate one of Britain’s greats, Greg Rutherford, rightly nominated this week in the twelve for BBC Sports Personality of the Year

Greg Rutherford’s fantastic long jump win at the World Championships meant he joined the select band of Brits holding the four major athletics titles at once. It was all the more fascinating because he has built a long- jump training pit in his back garden, as you can see below.

And a genuine GCSE Physics or Maths Higher tier question might be this: end of paper “tricky”, but in line with the emphasis on “real world problem solving”.

Question: Greg builds a long-jump run up and pit in his back garden. He typically accelerates evenly from 0 to 10 metres per second in 4 seconds, then runs for 2 more seconds at 10m/s before take off. The world record leap is then approximately 9 metres and he allows another 3 metres for landing. What is the minimum length Greg’s garden must be, from beginning of run up to end of landing?

Answer: in the first phase the word “evenly” implies a straight line velocity versus time graph from 0 to 4 seconds, and the distance covered is the area under that graph, namely half the base (2 seconds) times the height (10 m/s) i.e. 20m.

The Economist this week speculates that we are running out of combinations of letters for company names, and mentions the best and worst examples of made up names. One of the best is Google, which lead me to research its origin.

The good news is, there is a Maths angle.

The word Google comes from the googol, namely 10 to the power of 100, or 1 followed by one hundred zeros.

The founders of the company used the googol to represent the search engine idea of identifying an extremely large number of options. But the story goes that googol was
mis-spelled as google and the rest is history.

A nice GCSE question, in the new mode of “challenging”, might be:

A googol is 10 to the power 100

(a) What is a googol divided by ten to the power 98(b) Write in standard form 15 googols

These could be seen as frightening, yet easy at the same time:

(a) answer = 10² = 100
(b) answer 1.5 x ten to the power 101

The word googol itself was invented by a nine year old (why am I not surprised?) in the 1920’s. The nephew of American mathematician Edward Kasner. To get an idea of what a googol “looks like” it is similar to the ratio of the mass of an electron to the mass of the whole visible universe.

The word google in fact was mentioned before the company invention by an unlikely author, Enid Blyton. Not in “A very large number of people go the smuggler’s top” but in the term “Google Bun” in Faraway Magic Tree. Also (much more likely) Douglas Adams used the term Googleplex in the Hitchhiker’s Guide to the Galaxy, while Google itself uses “Googleplex” as the name for it’s HQ.

Googleplex is in fact the term for 10 to the power googol ( ten to the ten to the 100) which is a very large number indeed, perhaps to infinity and beyond. The mind boogles. I mean boggols. I mean boggles. in “Back to the Future 3″ the Doc says about future wife Clara ” She’s one in a billion. One in a Googleplex!”

The word googol surfaced again when it was the £1 million question in 2001 in Who Wants to Be a Millionaire?, the one where Charles Ingram was revealed to have used an accomplice.

Google (the word) is often in the news. It was the subject of an imaginary merger of the future with Amazon and subsequent war with Microsoft in (the Epic 2014 Googlezon wars).

It has officially become a verb (to Google, to search). Ironically Google the company doesn’t like this use, because it has come to mean “to search the whole web”, not just using their search engine, although most people do actually use Google as their primary search tool.

Google has been translated for instance into Chinese

After a financial reorganisation, Google the company name, has technically become “Alphabet” (a combination of word search and alpha-bet, the best algorithm choices). Personally I don’t think “Alphabet” will stick – the word will never catch on!

Finally, the Economist rated Google one of the best company names (becoming a verb clinched it). The worst? A large consultancy expensively renamed itself “Monday”, a name judged so bad that it did not last to the Friday, when it was taken over.

A headmaster in Wolverhampton has been suspended, and then reinstated after an enquiry, for allowing a pupil to take a GCSE English exam a day early. The reason seemed a little lax, namely to allow the pupil to go on holiday with their parents.

One assumes the enquiry involved checking his phone records and those of his friends in the few hours after the exam!

It reminds me of another “exam made easier” story from June when the answer to one GCSE question was helpfully supplied in another question, in the same paper. An AQA Chemistry paper contained the following:

2a. Fill in the blank. Limestone is mostly calcium ————
5b Limestone is made mostly of calcium carbonate…

In terms of making exams easier, let’s finish on a more serious note, well slightly more serious; allowing computers in exams.

The head of the OCR exam board suggests that Google be allowed in exams. The responses have varied from “ridiculous” and “rubbish” to “it would test resourcefulness and initiative rather than just your memory”.

Another proponent of the use of computers in exams is Dr. Sugata Mitra who conducted the famous experiment to place a computer in a hole in the wall adjacent to an Indian slum and found 7- year old children very quickly picked up skills with no assistance. It is a topic that won’t go away. But that is for another blog!

One Direction were in the news yesterday for postponing a concert, but the Mathemateer is more interested in the group’s Maths song. Yes there is one, and it’s good !
As described below – can you do the mental Maths? ! Hear it here.

One Direction were in the news earlier as they are rumoured to be taking a break in 2016. Recalling the excellent parody of their own song “What Makes You Beautiful” on Radio 1, the band wrote and recorded “Maths Song” whose main chorus is “Your Maths Skills = Terrible”

It features a series of quick, simple, mental maths tasks whose eventual answer is 130. At GCSE level this would not constitute a genuine question but on the other hand the lost art of mental maths should not be underestimated. As a warm up exercise before a test you could do worse than follow this through. Well done 1D as I think I should call them!

Another maths exam problem has gone viral after the earlier “sweets in a bag” Twitter storm. This time a Scottish Highers Maths question about crocodiles and zebras (yes!) proved insurmountable. Over and above the technical solution (see below) there were a number of interesting aspects for us English in GCSE – land.

First, the Scottish exam structure is completly different to England’s. There is no mention of GCSE or A-Level, so Higher in Scotland is roughly equivalent to A-Level in England, as it is described as a “pre-University qualification”.

Second, could such a question appear in English Maths GCSE ? Very unikely for the reason above, and because the best solution involves calculus, which is still not in the new GCSE 9-1 syallabus. Calculus is in IGCSE, but even so the crocodile problem would swallow up time as a very tricky differentiation is involved. It is, however, still just possible that a problem like this could be in our GCSE 9-1 syallabus because an alternate solution for it is through “iteration”. But solving it this way would surely eat up time, since perhaps 9 iterations might be needed with awkward square roots.

Third, it shows that quality control of questions is vital, especially when exam structures are changing. Ambiguity can be a killer. In this case many of the “descriptive” parts are not black and white (unlike the poor hunted zebra) . For instance how important is the width of the river? This makes even the first two “easy” parts tricky as you spend time understanding the English meaning. A shame – I feel the crocodile question writer (from Dundee?) crafted a potentially great question, but was let down at the end by the oversee process.

Fourth it shows there is a strong interest in Maths amongst the general public (I assume not crocodiles!) as the web post was No.1 in the charts for the BBC’s most read posts. This is encouraging!

Finally it shows there is no place that examiners won’t go to make questions less purely numeric, and more “challenging”. Another question involved toads and frogs down a well – let’s not go there.

For the record, the techncial solution (in summary!) is as follows – and it’s not a snappy answer!

(1) When x is 20, substituting in the equation for T gives T = 10.4 seconds
(2) When x is 0, substituting in the equation gives T = 11.0 seconds.

The minimum time T occurs when the dertivative (the differential) is equal to zero i.e. a turning point. Differentiating the equation and solving, we find x = 8. Substituting back in the original equation, we find that when x = 8, T = 9.8 seconds. We can prove that this turning point is a minimum by feeding in x values either side of 8 and showing that T is above 9.8 in both cases.

And that leads to the non-calculus “iteration” method, that in theory an English GCSE pupil could cope with. But you would have to start at x = 1, evaluate T, then use x = 2 and evaluate T again, and follow T down all the way down to x = 8, and find that T reduced to 9.8, Then for x = 9 find that T begins to increase again, i.e. a minimum had been reached at x = 8.

As I said, not snappy!

In conclusion, the question would not appear in GCSE south of the border due to content, and I don’t think it would make it to A-Level becasue of ambiguity – but its a salutory lesson for examiners.

England’s rugby team failed the test as the Wales match approached its climax. This has lessons for how to approach exams.

England exited the Rugby World Cup after losing to Australia, but for me the damage was done against Wales. I would argue that first loss was associated with “game management” and the parallels with “exam management” techniques are striking as we shall see.

A few weeks before the tournament, most critics would agree that the two teams were well matched. Then a few weeks before the match, Wales lost half their back line to injury, and during the match lost another half. Any small England selection errors were more than neutralised by Welsh misfortune. So duly, with 20 minutes to go, England were cruising, 10 points to the good, chances to extend. And yet they lost. Why?

In my view, the following: some bad luck with events, but mostly game management.

Harold MacMillan, former U.K. Prime Minister, once famously answered a journalist who had asked what could blow Governments off course: “events, dear boy, events”. England could not cope with events that should have been surmountable.

When Lloyd Williams hopefully kicked cross field, the oval ball could have bounced anywhere but in fact bounced perfectly into the hands of Gareth Davies to score. Bad luck, but it is how you react to events, and England’s game fell apart from there.

Another penalty conceded – more inability to understand what the referee wanted. (Are England penalised more than others despite, or because of, complaining a lot?) Then the fateful decision to go for the win instead of kicking for goal, not in itself illogical – the kick was missable, and risks sometimes are needed – but the decision to throw short at the line out, and risk being pushed into touch, was poor. Then one final chance, possession lost.

Stuart Lancaster, England coach, is reportedly a fan of the book and philosophy “The Score Takes Care of Itself” , in which Bill Walsh describes his experience as an American NFL Coach, arguing that the preparation, the little things, make the difference in leadership. Admirable, but no amount of preparation can overcome a coach or player’s inability to react to, or influence, events as they unfold.

Stuart is clearly a fantastic coach who oozes integrity, but before the tournament he said one thing which surprised me along the lines of, “my input to a match ceases just before it starts”. This refers of course to preparation, but did it betray an element of believing that events would follow the natural course, and so for instance substitutions would always follow at the preordained time?

You feel that New Zealand would also have taken the line out instead of the kick, but would have found a way to control it and win, borne of the confidence of winning late many times. They would have found a way to win.

The great sports people and teams keep their game management together as the pressure builds. Think, in contrast, of poor Jean van der Velde, the inexperienced French golfer who found himself only needing to avoid a triple bogey at the last to win the 1999 Open Championship at Carnoustie. In golf, we have “Course Management”, choosing the right clubs for distance, terrain, conditions. Unfortunately, Jean seemed to forget these guidelines, going via railings and rough to water. After removing socks and shoes he holed out for a 7 but lost the playoff,

England had one more chance, against Australia, but their confidence had gone. England lost the 1991 World Cup final against Australia because, by common consent, they listened to the critics and tried to play with flair instead of playing to their pack strength. Has the same thing happened recently? England have focussed on addressing their perceived weakness – the attacking flair – but judging by the way the Australian pack won scrum penalties, and had the edge at breakdown, it seems that England have let their forward advantage go.

And so England lost heavily to Australia. it probably would not have mattered had they beaten Wales. And that I would argue was due to Game Management.

Exam management

Think of the Maths exam as that rugby match. You are cruising through a twenty question paper, then just after half way you see a difficult question. You get stuck, you take too long. You begin to panic, you answer a question on “direct proportion” but forget the principle of feeding back the answer to double check. It turns out to be a wrong answer.

Then you see an algebraic question which requires a quadratic equation to be solved. However much you try, you cannot get the factors. But you haven’t noticed the question says “answer to 3 decimal points” (if so you would realise there are no factors as such, you have to use the quadratic formula).

Then a question involving Pi asks you to leave the answer “exact”, but instead you insert many of its ongoing figures rather than leaving Pi itself in. More marks lost. You fail to understand what the examiner wants, and what he will penalise you for.

You think you have an easy compound interest question. But you misread that it requires the final amount, not the interest paid. And you waste time doing the manual calculation as well, because you cannot find the “x to the n” button on the calculator.

A question on graphs is on the next page. You think, “this used to be my strength, but now It is all about Real Life Graphs, with wordy problems about bike rides and punctures. It’s a weakness now, I will have to pass!”

The next one looks easier. But no, it’s on Transformations. I can remember Reflections, but not the one that also begins with “Tr”? It all seemed so easy on my “maths-to-go” and “maths R us” websites. For a moment you remember an old black and white video you saw, what was his name, Brain Clough? “We had a good team on paper. Unfortunately football is played on grass”. You muse that this exam is the reverse, I can do the questions on the computer, unfortunately exams are on paper”.

Then finally you come to the last question. The bell will sound in a few minutes. It looks difficult but features probability, your strength. Decision time. Should I go back and pick up some easy marks by finishing an earlier one, or go for the five marker? You go for the latter.

But what’s this, it starts with probability and bags of sweets, and ends with an algebraic proof of n² – n – 90 = 0. “I have no clue how these things are connected! I give up”!

In conclusion

Could this happen, or is it just that nightmare where you dream you haven’t done your revision? Well consider this. It has happened and very recently. Thousands of students were approaching their Maths paper’s end – almost injury time so to speak – when they came across exactly that probability question above. The complaints caused a Twitter storm. Read the story, it went viral,

In fact a reasonable student could have solved this, had they stayed calm at the vital moment andlinked two seperate methods. Exam management, just like game and course management, can be the difference between achieving your goals and just missing out. You still made you’re A* to C, but not the A*. You have the abiility, but the sheer mechanics let you down.

The week between the Australia match and the final, irrelevant match against Uruguay will be the longest week of the team’s lives. Plenty of time to reflect on what might have been, just like the Summer Holidays for a student who might think “if only…”.

Connections between Maths and Music are many and varied. Here is another, indirectly at least. In Teddington the National Physical Laboratory and “Home of Measurement” plays host to the NPL Music Society, where small classical music lunchtime concerts are given in the Scientific Museum, Bushy House. These concerts feature pianists, singers, small chamber groups and recently a harpsichordist who perform in a room overlooking Bushy Park, while surrounded by all manner of scientific measuring instruments. The next performance is Thursday October 22nd 2015, featuring Haydn and Granados.

Meanwhile at Waldegrave School in Twickenham, a representative from the NPL recently gave a talk to the 6th Form Physics Group on the subject of standardised time zones and time measurement. Before the advent of the railways in the mid-19th century there were no standard time zones in the UK, and time differences between cities could vary by as much as 20 minutes, as explained in this article.

The NPL is home to the first Atomic Clock developed 60 years ago this year. The Caesium atomic clock is accurate to 1 second in 158 million years.

Maths GCSE includes questions on converting ratios with different units into “1 to n” ratios. It is an extreme example, but in this case the accuracy would be 1 to 158, 000,000 times the number of seconds in a year, which is 31,556, 926 (you didn’t know this? Nor did I!). Making : 4,982,688,000,000,000 in all, or about 1 in 5 million billion.

If you find that mind boggling consider this: the next generation of atomic clock will make the above look piffling, and will be 100 times more accurate, making an accuracy of 1 second in the age of the universe. I cannot get my head around that! It presumably would enable us to figure out if the Big Bang was late in coming, but that is another story, although Big Bang is actually covered in GCSE Science and Physics and also in Religious Studies. More on that another time.

Meanwhile back where we started, here is a link to an extensive review of a NPL concert from a couple of years ago and a more recent advert for a December 2015 concert featuring Natasha Hardy.

360 Degrees Of Billy Paul was one of the classic Philadelphia soul albums in the early 1970’s. It features the famous Gamble and Huff composition “Me and Mrs Jones”. Billy went on to record “Let Em In”, one of the few occasions, like Joe Cocker with a Little Help from My Friends, where the cover is arguably better than the original by a Beatle.

To be pedantic, Billy’s face only appears to be rotating 180°, nonetheless it is a classic album cover, and 360° features of course throughout GCSE Maths, in “bearings” questions, circular geometry, symmetry and segment analysis.

A typical foundation level question might be:

In the shape above, where is the line of symmetry? Answer is a line, drawn vertically down the middle.

Then a supplementary question about symmetry for higher level might be along the lines of:

If we then assume there is fourth hidden face at the back, and it is a 3-dimensional model, and you look down on it from the top, how many lines of symmetry are there? Answer: 4

And what is the order of rotational symmetry? Answer: 4 because there are 4 points through a rotation of 360° where the shape would look identical.

Watching The Life of Pi film again recently. Most people (OK 99.9%) of people remember the tiger, but the Mathemateer was most struck by the scene in which young Piscine Patel, tired of mockery, jumps up and announces his nickname is Pi, and what’s more can recite it to many decimal places. He writes 3.14159 …etc to many hundreds of decimal places on the board and achieves instant stardom.

Pi features in many GCSE Maths questions in formulae and it is really important for pupils to know which formulae are given in the formula sheet (for instance the volumes of spheres and cones) and which are not (quite rightly the formulae for a circle’s area and perimeter are not).

A typical higher level question might ask this:

In a full, tightly packed golf ball box there are two golf balls. What % of the volume of the box is occupied by the golf balls?

At first you think, we are not given any dimensions, how on earth can we solve this? The trick as you will increasingly see in the new syllabus is to think about a problem laterally and say, “Ok let’s call the radius r, see what happens, and start doing some calculations”. You will soon find that the volume of the box is 16 r³, while the volume of the two spheres is 8 pi r³/3 and a quick division gives you an answer of 52.4% because the “r” terms cancel out.

And finally, one more thing to remember. pi is an irrational number, which means it cannot be expressed as a whole number nor even a fraction. In fact it goes on forever, which is why Piscine is such a hero! And why it is used in “express to 4 significant figures” questions! Or why, if a GCSE question’s answer involves pi, and says “ leave as an exact answer” the pupil has to simply leave pi in the answer rather than try to work out the never ending, inexact, result. “Exam management” tips like this win points!